Dog owner here. I do sympathize with being grossed out by dog urine and feces if you’re going to set up on the beach with your kids. When I lived on the East Coast I loved that all the beaches are public and I loved running with my dog along the shore. There were doggie bag dispensers and trash cans everywhere so there’s no excuse to ever not pick up after your dog.
I don’t know what the answer is because some people won’t clean up after their dogs and give the rest of us a bad name. On the flip side... I’m grossed out by public pools because I know kids pee in the water and chlorine or not — I’m not sticking my head in that water. I also detest sidewalk spitters, sticky palm prints on restaurant doors, and sweaty people on gym workout equipment.
People will be gross. Their kids and dogs will do gross things. All you can do is try to be considerate yourself, appreciate people who do the same, and ignore things you can’t fix. Oh, and go gratifyingly postal when someone does something disgusting right in front of you. The latter can be really good for your system.
Not for me to say if it’s wrong. I say go for it, just don’t get yourself arrested.